Smoothness properties related to several commutators of fractional operators for values of $\boldsymbol{p}$ beyond the extreme in the multilinear setting

TL;DR

This paper investigates the continuity and smoothness properties of higher order commutators of multilinear fractional operators, extending known results to weighted spaces and broader parameter ranges.

Contribution

It provides new continuity results for multilinear fractional commutators on weighted Lebesgue and Lipschitz spaces, including a comprehensive study of multilinear weights and optimal parameter regions.

Findings

Extended estimates for unweighted and linear cases

Characterized multilinear weights and optimal parameter regions

Constructed examples of weights in the entire parameter space

Abstract

We prove continuity properties of higher order commutators of fractional operators on the multilinear setting, between a product of weighted Lebesgue spaces into certain weighted Lipschitz spaces. The considered operators include the multilinear fractional integral function and the main results extend previous estimates known for the unweighted case, as well as those established on the linear case. We give a complete study showing the main properties of the related multilinear weights and the optimal region described by the parameters where we can find nontrivial examples, including the restricted case of the one-weight theory. We also exhibit examples of such weights on the whole region.